Advertisements
Advertisements
Question
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Solution
x = 1 + 2i
∴ x – 1 = 2i
∴ (x – 1)2 = 4i2
∴ x2 – 2x + 1 = – 4 ...[∵ i2 = – 1]
∴ x2 – 2x + 5 = 0 ...(i)
x + 4
∵ `x^2 – 2x + 5")"overline(x^3 + 2x^2 - 3x + 21)"`
x3 – 2x2 + 5x
– + –
4x2 – 8x + 21
4x2 – 8x + 20
– + –
1
∴ x3 + 2x2 – 3x + 21
= (x2 – 2x + 5)(x + 4) + 1
= 0.(x + 4) + 1 ...[From (i)]
= 0 + 1
∴ x3 + 2x2 – 3x + 21 = 1
APPEARS IN
RELATED QUESTIONS
Find the value of i + i2 + i3 + i4
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Show that 1 + i10 + i100 − i1000 = 0
Evaluate : `("i"^37 + 1/"i"^67)`
Answer the following:
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
State true or false for the following:
If n is a positive integer, then the value of in + (i)n+1 + (i)n+2 + (i)n+3 is 0.
What is the value of `(i^(4n + 1) -i^(4n - 1))/2`?
What is the reciprocal of `3 + sqrt(7)i`.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
If z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is ______.
Simplify the following and express in the form a + ib.
`(3i^5 +2i^7 +i^9)/(i^6 +2i^8 +3i^18)`
